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Properties of Indices (Edexcel IGCSE Further Pure Maths)

Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Laws of Indices

What laws of indices do I need to know?

  • In an expression like a to the power of n 

    • a is known as the base

    • n is known as the index (also called the power or exponent)

  • The index laws you need to know and be able to use are summarised here:

    • a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent

    • a to the power of m divided by a to the power of n equals a to the power of m italic minus n end exponent

      • Be careful!  The two laws above can only be used if the two terms on the left-hand side of the equation have the same base.

    • stretchy left parenthesis a to the power of m stretchy right parenthesis to the power of n equals a to the power of m n end exponent

    • stretchy left parenthesis a b stretchy right parenthesis to the power of n equals a to the power of n b to the power of n

    • a to the power of 1 equals a

    • a to the power of 0 equals 1

    • a to the power of 1 over m end exponent equals m-th root of a

    • a to the power of m over n end exponent equals n-th root of a to the power of m end root equals open parentheses n-th root of a close parentheses to the power of m

    • a to the power of negative m end exponent equals 1 over a to the power of m

How do I work with laws of indices?

  • Laws of indices work with numerical and algebraic terms

  • These can be used to simplify expressions where terms are multiplied or divided

    • Deal with the number and algebraic parts separately

      • open parentheses 3 x to the power of 7 close parentheses cross times open parentheses 6 x to the power of 4 close parentheses equals open parentheses 3 cross times 6 close parentheses cross times open parentheses x to the power of 7 cross times x to the power of 4 close parentheses equals 18 x to the power of 11

      • fraction numerator 3 x to the power of 7 over denominator 6 x to the power of 4 end fraction equals 3 over 6 cross times x to the power of 7 over x to the power of 4 equals 1 half x cubed

      • open parentheses 3 x to the power of 7 close parentheses squared equals open parentheses 3 close parentheses squared cross times open parentheses x to the power of 7 close parentheses squared equals 9 x to the power of 14

How can I solve equations when the unknown is in the index?

  • If two terms with indices are equal and the terms have the same positive base (other than 1) then the indices must be equal

    • If a to the power of x equals a to the power of y then  x equals y

      • Not valid if a less or equal than 0 or a equals 1

  • If the unknown is part of the index then write both sides with the same base number

    • Then ignore the base number, make the indices equal and solve that equation

table row cell 5 to the power of 2 x end exponent end cell equals 125 row cell 5 to the power of 2 x end exponent end cell equals cell 5 cubed end cell row cell 2 x space end cell equals cell space 3 end cell row cell x space end cell equals cell space 3 over 2 end cell end table

  • In more complicated questions you might have to use negative and fractional indices

    • You may also have to rewrite both sides with the same base number

table row cell 8 to the power of x end cell equals cell 1 fourth end cell row cell open parentheses 2 cubed close parentheses to the power of x end cell equals cell 1 over 2 squared end cell row cell 2 to the power of 3 x end exponent end cell equals cell 2 to the power of negative 2 end exponent end cell row cell 3 x end cell equals cell negative 2 end cell row x equals cell negative 2 over 3 end cell end table

Worked Example

(a)table row cell blank to the power of blank end cell row blank end tableSimplify  fraction numerator left parenthesis 3 x squared right parenthesis left parenthesis 2 x cubed y squared right parenthesis over denominator left parenthesis 6 x squared y right parenthesis end fraction.


Multiply out the brackets in the numerator.
Rearrange the numerator so that you are multiplying the numbers together, the x terms together and the y terms together.

fraction numerator 3 cross times 2 cross times x squared cross times x cubed cross times y squared over denominator 6 x squared y end fraction

Simplify the numerator.
Multiply the constants together and add the powers of the x terms together.

fraction numerator 6 x to the power of 5 y squared over denominator 6 x squared y end fraction

Divide the constants.
Subtract the power of the x term in the denominator from the x term in the numerator: x to the power of 5 minus 2 end exponent equals x cubed.
Subtract the power of the y term in the denominator from the y term in the numerator: y to the power of 2 minus 1 end exponent equals y to the power of 1.

bold italic x to the power of bold 3 bold italic y

(b) Simplify  open parentheses fraction numerator 54 x to the power of 7 over denominator 2 x to the power of 4 end fraction close parentheses to the power of negative 1 third end exponent.

Simplify the expression inside the brackets.
Cancel down the constants.
Subtract the power of the x term in the denominator from the x term in the numerator: x to the power of 7 minus 4 end exponent equals x cubed.

open parentheses 27 x cubed close parentheses to the power of negative 1 third end exponent

Apply the negative index outside the brackets by 'flipping' the fraction inside the brackets.

open parentheses fraction numerator 1 over denominator 27 x cubed end fraction close parentheses to the power of 1 third end exponent

Apply the fractional index outside the brackets to everything inside the brackets.

fraction numerator 1 to the power of 1 third end exponent over denominator 27 to the power of 1 third end exponent x to the power of 3 cross times 1 third end exponent end fraction

Simplify.

fraction numerator bold 1 over denominator bold 3 bold italic x end fraction 

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Lucy Kirkham

    Author: Lucy Kirkham

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.